It is of course old in the art to provide machines for accomplishing a mechanical cutting of celery, carrots, and the like, and one popular machine of this general type utilizes a pair of endless moving belts disposed together so as to define an essentially V-shaped trough. Celery or carrots placed in such trough are carried one at a time through a cutting process, and although that type of arrangement is in widespread use, is fraught with numerous disadvantages. For example, that type of machine is limited to dealing with only limbs of a stalk of celery, so this of course requires that the limbs be stripped from the stalk before they are fed into that type of machine. There is a further requirement that the limbs be properly oriented as they are placed in the machine, which requires either additional complexity on the part of the machine, or else additional hand labor. As a result of these and other factors, it has in the past been quite expensive to slice elongate vegetables and this has, to a large extent, kept such products from being widely available on the shelves of grocery stores.